A dual finite element complex on the barycentric refinement

Given a two dimensional oriented surface equipped with a simplicial mesh, the standard lowest order finite element spaces provide a complex $X^\bullet$ centered on Raviart-Thomas divergence conforming vector fields. It can be seen as a realization of the simplicial cochain complex. We construct a new complex $Y^\bullet$ of finite element spaces on the barycentric refinement of the mesh which can be seen as a realization of the simplicial chain complex on the original (unrefined) mesh, such that the $\mathrm{L}^2$ duality is non-degenerate on $Y^i \times X^{2-i}$ for each $i\in \{0,1,2\}$. In particular $Y^1$ is a space of $\mathrm{curl}$-conforming vector fields which is $\mathrm{L}^2$ dual to Raviart-Thomas $\operatorname{div}$-conforming elements. When interpreted in terms of differential forms, these two complexes provide a finite-dimensional analogue of Hodge duality.